Reduced order modeling has gained considerable attention in recent decades owing to the advantages offered in reduced computational times and multiple solutions for parametric problems. The focus of this manuscript is the application of model order reduction techniques in various engineering and scientific applications including but not limited to mechanical, naval and aeronautical engineering. The focus here is kept limited to computational fluid mechanics and related applications. The advances in the reduced order modeling with proper orthogonal decomposition and reduced basis method are presented as well as a brief discussion of dynamic mode decomposition and also some present advances in the parameter space reduction. Here, an overview of the challenges faced and possible solutions are presented with examples from various problems.

}, url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85075395686\&partnerID=40\&md5=fb0b1a3cfdfd35a104db9921bc9be675}, author = {Gianluigi Rozza and M.H. Malik and Nicola Demo and Marco Tezzele and Michele Girfoglio and Giovanni Stabile and Andrea Mola} } @article {riccobelli2018activation, title = {Activation of a muscle as a mapping of stress{\textendash}strain curves}, journal = {Extreme Mech. Lett.}, volume = {28}, year = {2019}, pages = {37{\textendash}42}, publisher = {Elsevier BV}, author = {Davide Riccobelli and D. Ambrosi} } @article {20.500.11767_81643, title = {Accelerating the iterative solution of convection-diffusion problems using singular value decomposition}, journal = {NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS}, year = {2018}, pages = {1{\textendash}21}, doi = {10.1002/nla.2211}, url = {https://arxiv.org/abs/1807.09467}, author = {Giuseppe Pitton and Luca Heltai} } @article {MR3834681, title = {Adaptive discontinuous Galerkin methods for elliptic interface problems}, journal = {Math. Comp.}, volume = {87}, number = {314}, year = {2018}, pages = {2675{\textendash}2707}, issn = {0025-5718}, doi = {10.1090/mcom/3322}, url = {https://doi.org/10.1090/mcom/3322}, author = {Cangiani, Andrea and Georgoulis, Emmanuil H. and Sabawi, Younis A.} } @article {doi:10.1137/17M1147354, title = {Analysis of a Dynamic Peeling Test with Speed-Dependent Toughness}, journal = {SIAM Journal on Applied Mathematics}, volume = {78}, number = {2}, year = {2018}, pages = {1206-1227}, abstract = {We analyse a one-dimensional model of dynamic debonding for a thin film, where the local toughness of the glue between the film and the substrate also depends on the debonding speed. The wave equation on the debonded region is strongly coupled with Griffith{\textquoteright}s criterion for the evolution of the debonding front. We provide an existence and uniqueness result and find explicitly the solution in some concrete examples. We study the limit of solutions as inertia tends to zero, observing phases of unstable propagation, as well as time discontinuities, even though the toughness diverges at a limiting debonding speed.

}, doi = {10.1137/17M1147354}, url = {https://doi.org/10.1137/17M1147354}, author = {Giuliano Lazzaroni and Lorenzo Nardini} } @article {20.500.11767_81647, title = {An authenticated theoretical modeling of electrified fluid jet in core{\textendash}shell nanofibers production}, journal = {JOURNAL OF INDUSTRIAL TEXTILES}, volume = {47}, year = {2018}, pages = {1791{\textendash}1811}, doi = {10.1177/1528083717710711}, author = {Rafiei, S. and Noroozi, B. and Luca Heltai and Haghi, A. K.} } @article {2017, title = {Advances in Reduced order modelling for CFD: vortex shedding around a circular cylinder using a POD-Galerkin method}, journal = {Communication in Applied Industrial Mathematics}, year = {2017}, type = {reviewed}, abstract = {Vortex shedding around circular cylinders is a well known and studied phenomenon that appears in many engineering fields. In this work a Reduced Order Model (ROM) of the incompressible flow around a circular cylinder, built performing a Galerkin projection of the governing equations onto a lower dimensional space is presented. The reduced basis space is generated using a Proper Orthogonal Decomposition (POD) approach. In particular the focus is into (i) the correct reproduction of the pressure field, that in case of the vortex shedding phenomenon, is of primary importance for the calculation of the drag and lift coefficients; (ii) for this purpose the projection of the Governing equations (momentum equation and Poisson equation for pressure) is performed onto different reduced basis space for velocity and pressure, respectively; (iii) all the relevant modifications necessary to adapt standard finite element POD-Galerkin methods to a finite volume framework are presented. The accuracy of the reduced order model is assessed against full order results.

}, keywords = {finite volume, CFD, Reduced order methods}, url = {https://arxiv.org/abs/1701.03424}, author = {Giovanni Stabile and Saddam Hijazi and Stefano Lorenzi and Andrea Mola and Gianluigi Rozza} } @article {2017, title = {Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions}, number = {arXiv;1702.04674}, year = {2017}, abstract = {The goal of this monograph is to prove that any solution of the Cauchy problem for the capillarity-gravity water waves equations, in one space dimension, with periodic, even in space, initial data of small size ϵ, is almost globally defined in time on Sobolev spaces, i.e. it exists on a time interval of length of magnitude ϵ-N for any N, as soon as the initial data are smooth enough, and the gravity-capillarity parameters are taken outside an exceptional subset of zero measure. In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, our method is based on a normal forms procedure, in order to eliminate those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations are a quasi-linear system, usual normal forms approaches would face the well known problem of losses of derivatives in the unbounded transformations. In this monograph, to overcome such a difficulty, after a paralinearization of the capillarity-gravity water waves equations, necessary to obtain energy estimates, and thus local existence of the solutions, we first perform several paradifferential reductions of the equations to obtain a diagonal system with constant coefficients symbols, up to smoothing remainders. Then we may start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization.The reversible structure of the water waves equations, and the fact that we look for solutions even in x, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.}, url = {http://preprints.sissa.it/handle/1963/35285}, author = {Massimiliano Berti and Jean-Marc Delort} } @article {doi:10.1142/S2010326317400044, title = {Analytic geometry of semisimple coalescent Frobenius structures}, journal = {Random Matrices: Theory and Applications}, volume = {06}, number = {04}, year = {2017}, pages = {1740004}, abstract = {We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop {\textquotedblleft}Asymptotic and Computational Aspects of Complex Differential Equations{\textquotedblright} at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

}, doi = {10.1142/S2010326317400044}, url = {https://doi.org/10.1142/S2010326317400044}, author = {Giordano Cotti and Davide Guzzetti} } @article {feltrin2017, title = {An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators}, journal = {Topol. Methods Nonlinear Anal.}, volume = {50}, number = {2}, year = {2017}, pages = {683{\textendash}726}, publisher = {Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies}, doi = {10.12775/TMNA.2017.038}, url = {https://doi.org/10.12775/TMNA.2017.038}, author = {Guglielmo Feltrin and Fabio Zanolin} } @article {2017, title = {On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics}, journal = {Journal of Scientific Computing}, year = {2017}, abstract = {In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising.

}, doi = {10.1007/s10915-017-0419-6}, author = {Giuseppe Pitton and Gianluigi Rozza} } @article {FONDA20171064, title = {An avoiding cones condition for the Poincar{\'e}{\textendash}Birkhoff Theorem}, journal = {Journal of Differential Equations}, volume = {262}, number = {2}, year = {2017}, pages = {1064 - 1084}, abstract = {We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincar{\'e}{\textendash}Birkhoff fixed point Theorem for Poincar{\'e} maps of Hamiltonian systems.

}, keywords = {Avoiding cones condition, Hamiltonian systems, Periodic solutions, Poincar{\'e}{\textendash}Birkhoff theorem}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2016.10.002}, url = {http://www.sciencedirect.com/science/article/pii/S0022039616303278}, author = {Alessandro Fonda and Paolo Gidoni} } @article {MR3584583, title = {Adaptivity and blow-up detection for nonlinear evolution problems}, journal = {SIAM J. Sci. Comput.}, volume = {38}, number = {6}, year = {2016}, pages = {A3833{\textendash}A3856}, issn = {1064-8275}, doi = {10.1137/16M106073X}, url = {https://doi.org/10.1137/16M106073X}, author = {Cangiani, Andrea and Georgoulis, Emmanuil H. and Kyza, Irene and Metcalfe, Stephen} } @conference {2016, title = {Advances in geometrical parametrization and reduced order models and methods for computational fluid dynamics problems in applied sciences and engineering: overview and perspectives}, booktitle = {Proceedings of the ECCOMAS Congress 2016, VII European Conference on Computational Methods in Applied Sciences and Engineering,}, year = {2016}, month = {06/2016}, publisher = {ECCOMAS}, organization = {ECCOMAS}, address = {Crete, Greece}, abstract = {Several problems in applied sciences and engineering require reduction techniques in order to allow computational tools to be employed in the daily practice, especially in iterative procedures such as optimization or sensitivity analysis. Reduced order methods need to face increasingly complex problems in computational mechanics, especially into a multiphysics setting. Several issues should be faced: stability of the approximation, efficient treatment of nonlinearities, uniqueness or possible bifurcations of the state solutions, proper coupling between fields, as well as offline-online computing, computational savings and certification of errors as measure of accuracy. Moreover, efficient geometrical parametrization techniques should be devised to efficiently face shape optimization problems, as well as shape reconstruction and shape assimilation problems. A related aspect deals with the management of parametrized interfaces in multiphysics problems, such as fluid-structure interaction problems, and also a domain decomposition based approach for complex parametrized networks. We present some illustrative industrial and biomedical problems as examples of recent advances on methodological developments.

}, author = {Filippo Salmoiraghi and Francesco Ballarin and Giovanni Corsi and Andrea Mola and Marco Tezzele and Gianluigi Rozza}, editor = {Papadrakakis, M. and Papadopoulos, V. and Stefanou, G. and Plevris, V.} } @article {2013, title = {On the area of the graph of a piecewise smooth map from the plane to the plane with a curve discontinuity}, journal = {ESAIM: COCV}, volume = {22}, number = {arXiv:1310.2443;}, year = {2016}, pages = {29-63}, abstract = {In this paper we provide an estimate from above for the value of the relaxed area functional for a map defined on a bounded domain of the plane with values in the plane and discontinuous on a regular simple curve with two endpoints. We show that, under suitable assumptions, the relaxed area does not exceed the area of the regular part of the map, with the addition of a singular term measuring the area of a disk type solution of the Plateau{\textquoteright}s problem spanning the two traces of the map on the jump. The result is valid also when the area minimizing surface has self intersections. A key element in our argument is to show the existence of what we call a semicartesian parametrization of this surface, namely a conformal parametrization defined on a suitable parameter space, which is the identity in the first component. To prove our result, various tools of parametric minimal surface theory are used, as well as some result from Morse theory.

}, keywords = {Area functional}, doi = {10.1051/cocv/2014065}, url = {https://www.esaim-cocv.org/articles/cocv/abs/2016/01/cocv140065/cocv140065.html}, author = {Giovanni Bellettini and Lucia Tealdi and Maurizio Paolini} } @article {MR3589917, title = {On asymptotic regimes of orthogonal polynomials with complex varying quartic exponential weight}, journal = {SIGMA Symmetry Integrability Geom. Methods Appl.}, volume = {12}, year = {2016}, pages = {Paper No. 118, 50 pages}, issn = {1815-0659}, doi = {10.3842/SIGMA.2016.118}, url = {http://dx.doi.org/10.3842/SIGMA.2016.118}, author = {Marco Bertola and Alexander Tovbis} } @article {2015, title = {Anisotropic mean curvature on facets and relations with capillarity}, number = {Geometric Flows;1}, year = {2015}, publisher = {de Gruyter}, abstract = {We discuss the relations between the anisotropic calibrability of a facet F of a solid crystal E, and the capillary problem on a capillary tube with base F. When F is parallel to a facet of the Wulff shape, calibrability is equivalent to show the existence of an anisotropic subunitary vector field in $F, with suitable normal trace on the boundary of the facet, and with constant divergence equal to the anisotropic mean curvature of F. When the Wulff shape is a cylynder, assuming E convex at F, and F (strictly) calibrable, such a vector field is obtained by solving the capillary problem on F in absence of gravity and with zero contact angle. We show some examples of facets for which it is possible, even without the strict calibrability assumption, to build one of these vector fields. The construction provides, at least for convex facets of class C^{1,1}, the solution of the total variation flow starting at 1_F.

}, doi = {10.1515/geofl-2015-0005}, url = {http://urania.sissa.it/xmlui/handle/1963/34481}, author = {Stefano Amato and Lucia Tealdi and Giovanni Bellettini} } @article {MR3346719, title = {Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behavior and the first Painlev{\'e} equation}, journal = {Constr. Approx.}, volume = {41}, number = {3}, year = {2015}, pages = {529{\textendash}587}, issn = {0176-4276}, doi = {10.1007/s00365-015-9288-0}, url = {http://dx.doi.org/10.1007/s00365-015-9288-0}, author = {Marco Bertola and Alexander Tovbis} } @article {2014, title = {An Abstract Nash{\textendash}Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds}, number = {Communications in mathematical physics;volume 334; issue 3; pages 1413-1454;}, year = {2014}, publisher = {Springer}, abstract = {We prove an abstract implicit function theorem with parameters for smooth operators defined on scales of sequence spaces, modeled for the search of quasi-periodic solutions of PDEs. The tame estimates required for the inverse linearised operators at each step of the iterative scheme are deduced via a multiscale inductive argument. The Cantor-like set of parameters where the solution exists is defined in a non inductive way. This formulation completely decouples the iterative scheme from the measure theoretical analysis of the parameters where the small divisors non-resonance conditions are verified. As an application, we deduce the existence of quasi-periodic solutions for forced NLW and NLS equations on any compact Lie group or manifold which is homogeneous with respect to a compact Lie group, extending previous results valid only for tori. A basic tool of harmonic analysis is the highest weight theory for the irreducible representations of compact Lie groups.}, doi = {10.1007/s00220-014-2128-4}, url = {http://urania.sissa.it/xmlui/handle/1963/34651}, author = {Massimiliano Berti and Livia Corsi and Michela Procesi} } @article {2014, title = {Achieving unanimous opinions in signed social networks}, number = {2014 European Control Conference;article number 6862161; pages 184-189;}, year = {2014}, publisher = {Institute of Electrical and Electronics Engineers Inc.}, abstract = {Being able to predict the outcome of an opinion forming process is an important problem in social network theory. However, even for linear dynamics, this becomes a difficult task as soon as non-cooperative interactions are taken into account. Such interactions are naturally modeled as negative weights on the adjacency matrix of the social network. In this paper we show how the Perron-Frobenius theorem can be used for this task also beyond its standard formulation for cooperative systems. In particular we show how it is possible to associate the achievement of unanimous opinions with the existence of invariant cones properly contained in the positive orthant. These cases correspond to signed adjacency matrices having the eventual positivity property, i.e., such that in sufficiently high powers all negative entries have disappeared. More generally, we show how for social networks the achievement of a, possibily non-unanimous, opinion can be associated to the existence of an invariant cone fully contained in one of the orthants of n.}, doi = {10.1109/ECC.2014.6862161}, url = {http://urania.sissa.it/xmlui/handle/1963/34935}, author = {Claudio Altafini and Gabriele Lini} } @article {MR3269437, title = {Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems}, journal = {IMA J. Numer. Anal.}, volume = {34}, number = {4}, year = {2014}, pages = {1578{\textendash}1597}, issn = {0272-4979}, doi = {10.1093/imanum/drt052}, url = {https://doi.org/10.1093/imanum/drt052}, author = {Cangiani, Andrea and Georgoulis, Emmanuil H. and Metcalfe, Stephen} } @article {2014, title = {Adler-Gelfand-Dickey approach to classical W-algebras within the theory of Poisson vertex algebras}, number = {arXiv:1401.2082}, year = {2014}, note = {45 pages}, publisher = {SISSA}, abstract = {We put the Adler-Gelfand-Dickey approach to classical W-algebras in the framework of Poisson vertex algebras. We show how to recover the bi-Poisson structure of the KP hierarchy, together with its generalizations and reduction to the N-th KdV hierarchy, using the formal distribution calculus and the lambda-bracket formalism. We apply the Lenard-Magri scheme to prove integrability of the corresponding hierarchies. We also give a simple proof of a theorem of Kupershmidt and Wilson in this framework. Based on this approach, we generalize all these results to the matrix case. In particular, we find (non-local) bi-Poisson structures of the matrix KP and the matrix N-th KdV hierarchies, and we prove integrability of the N-th matrix KdV hierarchy.}, url = {http://hdl.handle.net/1963/7242}, author = {Alberto De Sole and Victor G. Kac and Daniele Valeri} } @article {2014, title = {Approximate Hermitian{\textendash}Yang{\textendash}Mills structures on semistable principal Higgs bundles}, number = {Annals of global analysis and geometry;volume 47; issue 1; pp 1-11}, year = {2014}, publisher = {Springer}, abstract = {We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.}, doi = {10.1007/s10455-014-9433-1}, url = {http://urania.sissa.it/xmlui/handle/1963/34645}, author = {Ugo Bruzzo and Beatriz Grana-Otero} } @article {2014, title = {Approximate Hitchin-Kobayashi correspondence for Higgs G-bundles}, number = {International Journal of Geometric Methods in Modern Physics;volume 11; issue 7; article number 1460015;}, year = {2014}, publisher = {World Scientific Publishing}, abstract = {We announce a result about the extension of the Hitchin-Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact K{\"a}hler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.}, doi = {10.1142/S0219887814600159}, url = {http://urania.sissa.it/xmlui/handle/1963/35095}, author = {Ugo Bruzzo and Beatriz Gra{\~n}a Otero} } @article {2013, title = {Ambrosio-Tortorelli approximation of cohesive fracture models in linearized elasticity}, year = {2013}, institution = {SISSA}, abstract = {We provide an approximation result in the sense of $\Gamma$-convergence for cohesive fracture energies of the form \[ \int_\Omega \mathscr{Q}_1(e(u))\,dx+a\,\mathcal{H}^{n-1}(J_u)+b\,\int_{J_u}\mathscr{Q}_0^{1/2}([u]\odot\nu_u)\,d\mathcal{H}^{n-1}, \] where $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary, $\mathscr{Q}_0$ and $\mathscr{Q}_1$ are coercive quadratic forms on ${\mathbb M}^{n\times n}_{sym}$, $a,\,b$ are positive constants, and $u$ runs in the space of fields $SBD^2(\Omega)$ , i.e., it{\textquoteright}s a special field with bounded deformation such that its symmetric gradient $e(u)$ is square integrable, and its jump set $J_u$ has finite $(n-1)$-Hausdorff measure in ${\mathbb R}^n$. The approximation is performed by means of Ambrosio-Tortorelli type elliptic regularizations, the prototype example being \[ \int_\Omega\Big(v|e(u)|^2+\frac{(1-v)^2}{\varepsilon}+{\gamma\,\varepsilon}|\nabla v|^2\Big)\,dx, \] where $(u,v)\in H^1(\Omega,{\mathbb R}^n){\times} H^1(\Omega)$, $\varepsilon\leq v\leq 1$ and $\gamma\>0$.

}, keywords = {Functions of bounded deformation}, url = {http://hdl.handle.net/1963/6615}, author = {Matteo Focardi and Flaviana Iurlano} } @article {2013, title = {Analytical validation of a continuum model for epitaxial growth with elasticity on vicinal surfaces}, number = {Archive for Rational Mechanics and Analysis}, year = {2013}, publisher = {Springer}, abstract = {In this paper it is shown existence of weak solutions of a variational inequality derived from the continuum model introduced by Xiang [7, formula (3.62)] (see also the work of Xiang and E [8] and Xu and Xiang [9]) to describe the self-organization of terraces and steps driven by misfit elasticity between a film and a substrate in heteroepitaxial growth. This model is obtained as a continuum limit of discrete theories of Duport, Politi, and Villain [3] and Tersoff, Phang, Zhang, and Lagally[6].}, keywords = {singular nonlinear parabolic equations, Hilbert transform, thin films}, doi = {10.1007/s00205-014-0730-4}, url = {http://hdl.handle.net/1963/7245}, author = {Gianni Dal Maso and Irene Fonseca and Giovanni Leoni} } @mastersthesis {2013, title = {An Approximation Result for Generalised Functions of Bounded Deformation and Applications to Damage Problems}, year = {2013}, school = {SISSA}, keywords = {Functions of bounded deformation}, author = {Flaviana Iurlano} } @article {2013, title = {Asymptotics of the first Laplace eigenvalue with Dirichlet regions of prescribed length}, number = {SIAM Journal on Mathematical Analysis;volume 45; issue 6; pp. 3266-3282}, year = {2013}, publisher = {Society for Industrial and Applied Mathematics}, abstract = {We consider the problem of maximizing the first eigenvalue of the $p$-Laplacian (possibly with nonconstant coefficients) over a fixed domain $\Omega$, with Dirichlet conditions along $\partial\Omega$ and along a supplementary set $\Sigma$, which is the unknown of the optimization problem. The set $\Sigma$, which plays the role of a supplementary stiffening rib for a membrane $\Omega$, is a compact connected set (e.g., a curve or a connected system of curves) that can be placed anywhere in $\overline{\Omega}$ and is subject to the constraint of an upper bound $L$ to its total length (one-dimensional Hausdorff measure). This upper bound prevents $\Sigma$ from spreading throughout $\Omega$ and makes the problem well-posed. We investigate the behavior of optimal sets $\Sigma_L$ as $L\to\infty$ via $\Gamma$-convergence, and we explicitly construct certain asymptotically optimal configurations. We also study the behavior as $p\to\infty$ with $L$ fixed, finding connections with maximum-distance problems related to the principal frequency of the $\infty$-Laplacian.}, doi = {10.1137/130916825}, url = {http://urania.sissa.it/xmlui/handle/1963/35141}, author = {Paolo Tilli and Davide Zucco} } @article {2013, title = {Attainment results for nematic elastomers}, year = {2013}, institution = {SISSA}, abstract = {We consider a class of non-quasiconvex frame indifferent energy densities which includes Ogden-type energy densities for nematic elastomers. For the corresponding geometrically linear problem we provide an explicit minimizer of the energy functional satisfying a nontrivial boundary condition. Other attainment results, both for the nonlinear and the linearized model, are obtained by using the theory of convex integration introduced by Mueller and Sverak in the context of crystalline solids.}, url = {http://hdl.handle.net/1963/7174}, author = {Virginia Agostiniani and Gianni Dal Maso and Antonio DeSimone} } @article {2012, title = {Asymptotics of the s-perimeter as s {\textrightarrow}0 }, journal = {Discrete Contin. Dyn. Syst. 33, nr.7 (2012): 2777-2790}, number = {arXiv:1204.0750v2;}, year = {2012}, publisher = {American Institute of Mathematical Sciences}, abstract = {We deal with the asymptotic behavior of the $s$-perimeter of a set $E$ inside a domain $\Omega$ as $s\searrow0$. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of $E$ and $\Omega$. Moreover, we construct examples of sets for which the limit does not exist.

}, doi = {10.3934/dcds.2013.33.2777}, author = {Serena Dipierro and Alessio Figalli and Giampiero Palatucci and Enrico Valdinoci} } @article {2011, title = {Adaptation as a genome-wide autoregulatory principle in the stress response of yeast.}, journal = {IET systems biology. 2011 Jul; 5(4):269-79}, number = {PMID:21823758;}, year = {2011}, publisher = {The Institution of Engineering and Technology}, abstract = {The gene expression response of yeast to various types of stresses/perturbations shows a common functional and dynamical pattern for the vast majority of genes, characterised by a quick transient peak (affecting primarily short genes) followed by a return to the pre-stimulus level. Kinetically, this process of adaptation following the transient excursion can be modelled using a genome-wide autoregulatory mechanism by means of which yeast aims at maintaining a preferential concentration in its mRNA levels. The resulting feedback system explains well the different time constants observable in the transient response, while being in agreement with all the known experimental dynamical features. For example, it suggests that a very rapid transient can be induced also by a slowly varying concentration of the gene products.}, doi = {10.1049/iet-syb.2009.0050}, url = {http://hdl.handle.net/1963/5106}, author = {F Eduati and B Di Camillo and G Toffolo and Claudio Altafini and Giovanna De Palo and Mattia Zampieri} } @article {2011, title = {An asymptotic reduction of a Painlev{\'e} VI equation to a Painlev{\'e} III}, journal = {J.Phys.A: Math.Theor. 44 (2011) 215203}, number = {arXiv:1101.4705;}, year = {2011}, publisher = {IOP Publishing}, abstract = {When the independent variable is close to a critical point, it is shown that\\r\\nPVI can be asymptotically reduced to PIII. In this way, it is possible to\\r\\ncompute the leading term of the critical behaviors of PVI transcendents\\r\\nstarting from the behaviors of PIII transcendents.}, doi = {10.1088/1751-8113/44/21/215203}, url = {http://hdl.handle.net/1963/5124}, author = {Davide Guzzetti} } @article {2011, title = {Axial symmetry of some steady state solutions to nonlinear Schr{\"o}dinger equations}, journal = {Proc. Amer. Math. Soc. 139 (2011), 1023-1032}, number = {SISSA;75/2010/M}, year = {2011}, publisher = {American Mathematical Society}, abstract = {In this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space.}, keywords = {Nonlinear Schr{\"o}dinger equation}, doi = {10.1090/S0002-9939-2010-10638-X}, url = {http://hdl.handle.net/1963/4100}, author = {Changfeng Gui and Andrea Malchiodi and Haoyuan Xu and Paul Yang} } @article {Berti2010377, title = {An abstract Nash-Moser theorem with parameters and applications to PDEs}, journal = {Annales de l{\textquoteright}Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis}, volume = {27}, number = {1}, year = {2010}, note = {cited By (since 1996)9}, pages = {377-399}, abstract = {We prove an abstract Nash-Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the "tame" estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large "clusters of small divisors", due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity. {\textcopyright} 2009 Elsevier Masson SAS. All rights reserved.}, keywords = {Abstracting, Aircraft engines, Finite dimensional, Hamiltonian PDEs, Implicit function theorem, Invariant tori, Iterative schemes, Linearized operators, Mathematical operators, Moser theorem, Non-Linearity, Nonlinear equations, Nonlinear wave equation, Periodic solution, Point of interest, Resonance phenomena, Small divisors, Sobolev, Wave equations}, issn = {02941449}, doi = {10.1016/j.anihpc.2009.11.010}, author = {Massimiliano Berti and Philippe Bolle and Michela Procesi} } @mastersthesis {2010, title = {Almost-Riemannian Geometry from a Control Theoretical Viewpoint}, year = {2010}, school = {SISSA}, url = {http://hdl.handle.net/1963/4705}, author = {Roberta Ghezzi} } @proceedings {2010, title = {Aspects of Quantum Field Theory on Quantum Spacetime}, number = {arXiv:1103.3405v1;}, year = {2010}, note = {25 pages, active hyperlinks. Corfu Summer Institute on Elementary\\r\\n Particles and Physics - Workshop on Non Commutative Field Theory and Gravity,\\r\\n September 8-12, 2010, Corfu Greece}, publisher = {SISSA}, abstract = {We provide a minimal, self-contained introduction to the covariant DFR flat\\r\\nquantum spacetime, and to some partial results for the corresponding quantum field theory. Explicit equations are given in the Dirac notation.}, url = {http://hdl.handle.net/1963/4171}, author = {Gherardo Piacitelli} } @article {SELVITELLA20082566, title = {Asymptotic evolution for the semiclassical nonlinear Schr{\"o}dinger equation in presence of electric and magnetic fields}, journal = {Journal of Differential Equations}, volume = {245}, number = {9}, year = {2008}, pages = {2566 - 2584}, abstract = {In this paper we study the semiclassical limit for the solutions of a subcritical focusing NLS with electric and magnetic potentials. We consider in particular the Cauchy problem for initial data close to solitons and show that, when the Planck constant goes to zero, the motion shadows that of a classical particle. Several works were devoted to the case of standing waves: differently from these we show that, in the dynamic version, the Lorentz force appears crucially.

}, issn = {0022-0396}, doi = {https://doi.org/10.1016/j.jde.2008.05.012}, url = {http://www.sciencedirect.com/science/article/pii/S002203960800243X}, author = {Alessandro Selvitella} } @article {2007, title = {Asymptotic behaviour of smooth solutions for partially dissipative hyperbolic systems with a convex entropy}, journal = {Comm. Pure Appl. Math. 60 (2007) 1559-1622}, number = {SISSA;83/2005/M}, year = {2007}, abstract = {We study the asymptotic time behavior of global smooth solutions to general entropy dissipative hyperbolic systems of balance law in m space dimensions, under the Shizuta-Kawashima condition.}, doi = {10.1002/cpa.20195}, url = {http://hdl.handle.net/1963/1780}, author = {Stefano Bianchini and Bernard Hanouzet and Roberto Natalini} } @article {2007, title = {The Asymptotic Behaviour of the Fourier Transforms of Orthogonal Polynomials II: L.I.F.S. Measures and Quantum Mechanics}, journal = {Ann. Henri Poincar{\textasciiacute}e 8 (2007), 301{\textendash}336}, year = {2007}, publisher = {2007 Birkh{\textasciidieresis}auser Verlag Basel/Switzerland}, abstract = {We study measures generated by systems of linear iterated functions,\r\ntheir Fourier transforms, and those of their orthogonal polynomials. We\r\ncharacterize the asymptotic behaviours of their discrete and continuous averages.\r\nFurther related quantities are analyzed, and relevance of this analysis\r\nto quantum mechanics is briefly discussed}, doi = {10.1007/s00023-006-0309-1}, author = {Davide Guzzetti and Giorgio Mantica} } @article {2007, title = {Asymptotic variational wave equations}, journal = {Arch. Ration. Mech. Anal. 183 (2007) 163-185}, number = {arXiv.org;math/0502124v1}, year = {2007}, abstract = {We investigate the equation $(u_t + (f(u))_x)_x = f\\\'\\\'(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.\\nWe establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data.}, doi = {10.1007/s00205-006-0014-8}, url = {http://hdl.handle.net/1963/2182}, author = {Alberto Bressan and Zhang Ping and Zheng Yuxi} } @article {2006, title = {Almost Global Stochastic Feedback Stabilization of Conditional Quantum Dynamics}, number = {SISSA;80/2005/M}, year = {2006}, abstract = {We propose several parametrization-free solutions to the problem of quantum state reduction control by means of continuous measurement and smooth quantum feedback. In particular, we design a feedback law for which almost global stochastic feedback stabilization can be proved analytically by means of Lyapunov techinques. This synthesis arises very naturally from the physics of the problem, as it relies on the variance associated with the quantum filtering process.}, url = {http://hdl.handle.net/1963/1727}, author = {Claudio Altafini and Francesco Ticozzi} } @article {2006, title = {An artificial viscosity approach to quasistatic crack growth}, number = {SISSA;43/2006/M}, year = {2006}, abstract = {We introduce a new model of irreversible quasistatic crack growth in which the evolution of cracks is the limit of a suitably modified $\\\\epsilon$-gradient flow of the energy functional, as the \\\"viscosity\\\" parameter $\\\\epsilon$ tends to zero.}, url = {http://hdl.handle.net/1963/1850}, author = {Rodica Toader and Chiara Zanini} } @article {2005, title = {Asymptotic Morse theory for the equation $\\\\Delta v=2v\\\\sb x\\\\wedge v\\\\sb y$}, journal = {Comm. Anal. Geom. 13 (2005) 187-252}, number = {arXiv.org;math/0205106v1}, year = {2005}, publisher = {International Press}, abstract = {Given a smooth bounded domain ${\\\\O}\\\\subseteq \\\\R^2$, we consider the equation $\\\\D v = 2 v_x \\\\wedge v_y$ in $\\\\O$, where $v: {\\\\O}\\\\to \\\\R^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple-bubbling phenomena. This allows us to settle a particular case of a question raised by H. Brezis and J.M. Coron.}, url = {http://hdl.handle.net/1963/3533}, author = {Sagun Chanillo and Andrea Malchiodi} } @article {2005, title = {On the attainable set for Temple class systems with boundary controls}, journal = {SIAM J. Control Optim. 43 (2005) 2166-2190}, number = {SISSA;10/2002/M}, year = {2005}, publisher = {SISSA Library}, abstract = {Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws \% $$ u_t+f(u)_x=0, \\\\qquad u(0,x)=\\\\ov u(x), \\\\qquad {{array}{ll} \&u(t,a)=\\\\widetilde u_a(t), \\\\noalign{\\\\smallskip} \&u(t,b)=\\\\widetilde u_b(t), {array}. \\\\eqno(1) $$ on the domain $\\\\Omega =\\\\{(t,x)\\\\in\\\\R^2 : t\\\\geq 0, a \\\\le x\\\\leq b\\\\}.$ We study the mixed problem (1) from the point of view of control theory, taking the initial data $\\\\bar u$ fixed, and regarding the boundary data $\\\\widetilde u_a, \\\\widetilde u_b$ as control functions that vary in prescribed sets $\\\\U_a, \\\\U_b$, of $\\\\li$ boundary controls. In particular, we consider the family of configurations $$ \\\\A(T) \\\\doteq \\\\big\\\\{u(T,\\\\cdot); ~ u {\\\\rm is a sol. to} (1), \\\\quad \\\\widetilde u_a\\\\in \\\\U_a, \\\\widetilde u_b \\\\in \\\\U_b \\\\big\\\\} $$ that can be attained by the system at a given time $T>0$, and we give a description of the attainable set $\\\\A(T)$ in terms of suitable Oleinik-type conditions. We also establish closure and compactness of the set $\\\\A(T)$ in the $lu$ topology.}, doi = {10.1137/S0363012902407776}, url = {http://hdl.handle.net/1963/1581}, author = {Fabio Ancona and Giuseppe Maria Coclite} } @article {2004, title = {On almost duality for Frobenius manifolds}, journal = {Amer. Math. Soc. Transl. 212 (2004)\\n75-132.}, number = {arXiv.org;math/0307374}, year = {2004}, abstract = {We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg - Witten duality.}, url = {http://hdl.handle.net/1963/2543}, author = {Boris Dubrovin} } @article {2004, title = {On analytic families of invariant tori for PDEs}, journal = {Ast{\'e}risque. Issue 297, 2004, Pages 35-65}, year = {2004}, publisher = {SISSA}, abstract = {We propose to apply a version of the classical Stokes\\r\\nexpansion method to the perturbative construction of invariant tori for\\r\\nPDEs corresponding to solutions quasiperiodic in space and time variables.\\r\\nWe argue that, for integrable PDEs all but finite number of the\\r\\nsmall divisors arising in the perturbative analysis cancel. As an illustrative\\r\\nexample we establish such cancellations for the case of KP equation.\\r\\nIt is proved that, under mild assumptions about decay of the magnitude\\r\\nof the Fourier modes all analytic families of finite-dimensional invariant\\r\\ntori for KP are given by the Krichever construction in terms of thetafunctions\\r\\nof Riemann surfaces. We also present an explicit construction\\r\\nof infinite dimensional real theta-functions and corresponding quasiperiodic\\r\\nsolutions to KP as sums of infinite number of interacting plane\\r\\nwaves.}, url = {http://hdl.handle.net/1963/6474}, author = {Boris Dubrovin} } @article {2004, title = {Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains}, journal = {Ann. Inst. H. Poincar{\'e}. Anal. Non Lin{\'e}aire 21 (2004), (4), p. 445-486.}, number = {SISSA;40/2002/M}, year = {2004}, publisher = {SISSA Library}, abstract = {We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_0) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H-converge to a matrix A^0, we prove that there exist a subsequence and a measure mu^0 in M_0 such that the limit problem is the relaxed Dirichlet problem corresponding to A^0 and mu^0. We also prove a corrector result which provides an explicit approximation of the solutions in the H^1-norm, and which is obtained by multiplying the corrector for the H-converging matrices by some special test function which depends both on the varying matrices and on the varying domains.}, url = {http://hdl.handle.net/1963/1611}, author = {Gianni Dal Maso and Francois Murat} } @article {2003, title = {Autonomous integral functionals with discontinous nonconvex integrands: Lipschitz regularity of mimimizers, DuBois-Reymond necessary conditions and Hamilton-Jacobi equations}, journal = {Applied Math.Optim. 48 (2003), no.1, p.39-66}, number = {SISSA;55/2002/M}, year = {2003}, publisher = {SISSA Library}, doi = {10.1007/s00245-003-0768-4}, url = {http://hdl.handle.net/1963/1625}, author = {Gianni Dal Maso and Helene Frankowska} } @article {2002, title = {Admissible Riemann solvers for genuinely nonlinear P-systems of mixed type}, journal = {J. Differ. Equations, 2002, 180, 395}, number = {SISSA;33/00/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1006/jdeq.2001.4066}, url = {http://hdl.handle.net/1963/1491}, author = {Jean-Marc Mercier and Benedetto Piccoli} } @article {2002, title = {Arnold diffusion: a functional analysis approach}, journal = {Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsional. Akad. Nauk Ukra{\"\i}ni, Inst. Mat., Kiev, 2002}, year = {2002}, publisher = {Natsional. Akad. Nauk Ukra{\"\i}ni}, abstract = {We present, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to study the {\textquotedblleft}splitting of the whiskers{\textquotedblright} and the {\textquotedblleft}shadowing problem{\textquotedblright} developed in collaboration with P. Bolle in the recent papers [1] and [2] . This method is applied to the problem of Arnold diffusion for nearly integrable partially isochronous systems improving known results.}, author = {Massimiliano Berti} } @article {2001, title = {Adiabatic limits of closed orbits for some Newtonian systems in R-n}, journal = {Asymptotic Anal., 2001, 25, 149-181}, number = {SISSA;53/00/M}, year = {2001}, publisher = {SISSA Library}, abstract = {We deal with a Newtonian system like x\\\'\\\' + V\\\'(x) = 0. We suppose that V: \\\\R^n \\\\to \\\\R possesses an (n-1)-dimensional compact manifold M of critical points, and we prove the existence of arbitrarity slow periodic orbits. When the period tends to infinity these orbits, rescaled in time, converge to some closed geodesics on M.}, url = {http://hdl.handle.net/1963/1511}, author = {Andrea Malchiodi} } @article {2000, title = {Abnormal extremals for minimum time on the plane}, number = {SISSA;50/00/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1508}, author = {Ugo Boscain and Benedetto Piccoli} } @article {2000, title = {Arnold{\textquoteright}s Diffusion in nearly integrable isochronous Hamiltonian systems}, number = {SISSA;98/00/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1554}, author = {Massimiliano Berti and Philippe Bolle} } @article {2000, title = {A(SLq(2)) at roots of unity is a free module over A(SL(2))}, journal = {Lett. Math. Phys., 2000, 52, 339}, number = {SISSA;42/00/FM}, year = {2000}, publisher = {SISSA Library}, doi = {10.1023/A:1007601131002}, url = {http://hdl.handle.net/1963/1500}, author = {Ludwik Dabrowski and Cesare Reina and Alessandro Zampa} } @article {1999, title = {The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional}, journal = {Numer. Funct. Anal. Optim. 20 (1999), no. 9-10, 957-982}, number = {SISSA;62/99/M}, year = {1999}, publisher = {Taylor and Francis}, abstract = {We compute explicitly the anisotropy effect in the H1 term, generated in the approximation of the Mumford-Shah functional by finite element spaces defined on structured triangulations.}, doi = {10.1080/01630569908816934}, url = {http://hdl.handle.net/1963/1276}, author = {Matteo Negri} } @mastersthesis {1999, title = {Approximation, Stability and control for Conservation Laws}, year = {1999}, school = {SISSA}, url = {http://hdl.handle.net/1963/5500}, author = {Andrea Marson} } @article {1999, title = {Asymptotic behaviour of nonlinear elliptic higher order equations in perforated domains}, journal = {Journal d\\\'Analyse Mathematique, Volume 79, 1999, Pages: 63-112}, year = {1999}, isbn = {1618-1891}, doi = {10.1007/BF01759365}, url = {http://hdl.handle.net/1963/6433}, author = {Gianni Dal Maso and Igor V. Skrypnik} } @mastersthesis {1998, title = {Algebraic Solutions to the Painlev{\'e}-VI Equation and Reflection Groups}, year = {1998}, school = {SISSA}, keywords = {Painlev{\'e} VI equation}, url = {http://hdl.handle.net/1963/5574}, author = {Marta Mazzocco} } @article {1998, title = {Asymptotic behavior of nonlinear Dirichlet problems in perforated domains}, journal = {Ann. Mat. Pura Appl. (4) 174 (1998), 13--72}, number = {SISSA;162/95/M}, year = {1998}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1064}, author = {Gianni Dal Maso and Igor V. Skrypnik} } @article {1994, title = {Algebraic-geometrical Darboux coordinates in R-matrix formalism}, number = {SISSA;88/1994/FM}, year = {1994}, institution = {SISSA}, url = {http://hdl.handle.net/1963/3655}, author = {P. Diener and Boris Dubrovin} } @mastersthesis {1994, title = {Analysis of Singularity Structures for Quasi-Integrable Hamiltonian Systems}, year = {1994}, school = {SISSA}, keywords = {Hamiltonian systems}, url = {http://hdl.handle.net/1963/5685}, author = {Simonetta Abenda} } @mastersthesis {1994, title = {Asymptotic Behaviour of Dirichlet Problems in Perforated Domains}, year = {1994}, school = {SISSA}, keywords = {Dirichlet problems}, url = {http://hdl.handle.net/1963/5714}, author = {Adriana Garroni} } @article {1990, title = {Algebraic differential calculus for gauge theories}, journal = {Nuclear Phys. B. Proc. Suppl. 18A (1990), 171}, number = {SISSA;135/89/FM}, year = {1990}, publisher = {SISSA Library}, doi = {10.1016/0920-5632(90)90649-F}, url = {http://hdl.handle.net/1963/891}, author = {Giovanni Landi and Giuseppe Marmo} } @article {1989, title = {An approach to the thin obstacle problem for variational functionals depending on vector}, journal = {Comm. Partial Differential Equations, 14 (1989), no.12, 1717-1743.}, number = {SISSA;41/89/M}, year = {1989}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/802}, author = {Gianni Dal Maso and Roberta Musina} } @article {1988, title = {Algebraic reduction of the \\\'t Hooft-Polyakov monopole to the Dirac monopole.}, journal = {Phys. Lett. B 201 (1988), no. 1, 101-104.}, number = {SISSA;97/87/FM}, year = {1988}, publisher = {SISSA Library}, doi = {10.1016/0370-2693(88)90088-3}, url = {http://hdl.handle.net/1963/578}, author = {Giovanni Landi and Giuseppe Marmo} } @mastersthesis {1988, title = {An Algebraic Setting for Gauge Theories}, year = {1988}, school = {SISSA}, url = {http://hdl.handle.net/1963/5828}, author = {Giovanni Landi} } @article {1983, title = {On the asymptotic behaviour of solutions to Pazy\\\'s class of evolution equations}, number = {SISSA;37/83/M}, year = {1983}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/276}, author = {Giovanni Vidossich} }